LMFDB - Embedded newform 1120.2.n.b.559.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,2,Mod(559,1120)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1120.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.n (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.4
Character	$\chi$	$=$	1120.559
Dual form			1120.2.n.b.559.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.91053 q^{3} +(-1.83654 - 1.27559i) q^{5} +(-1.52252 + 2.16378i) q^{7} +5.47117 q^{9} +O(q^{10})$	\(q-2.91053 q^{3} +(-1.83654 - 1.27559i) q^{5} +(-1.52252 + 2.16378i) q^{7} +5.47117 q^{9} +0.0929484 q^{11} +4.08979i q^{13} +(5.34529 + 3.71264i) q^{15} -4.24008 q^{17} +2.39835i q^{19} +(4.43134 - 6.29773i) q^{21} -4.52148 q^{23} +(1.74573 + 4.68534i) q^{25} -7.19241 q^{27} +4.35133i q^{29} +1.10465 q^{31} -0.270529 q^{33} +(5.55626 - 2.03174i) q^{35} +8.54612 q^{37} -11.9034i q^{39} -6.10466i q^{41} -4.60415i q^{43} +(-10.0480 - 6.97898i) q^{45} -7.93489i q^{47} +(-2.36386 - 6.58879i) q^{49} +12.3409 q^{51} -14.3740 q^{53} +(-0.170703 - 0.118564i) q^{55} -6.98047i q^{57} -10.6226i q^{59} -1.92656 q^{61} +(-8.32997 + 11.8384i) q^{63} +(5.21690 - 7.51105i) q^{65} -13.1636i q^{67} +13.1599 q^{69} +9.75994i q^{71} +6.19245 q^{73} +(-5.08100 - 13.6368i) q^{75} +(-0.141516 + 0.201119i) q^{77} +3.42089i q^{79} +4.52019 q^{81} +11.7848 q^{83} +(7.78705 + 5.40860i) q^{85} -12.6647i q^{87} -4.46220i q^{89} +(-8.84939 - 6.22679i) q^{91} -3.21511 q^{93} +(3.05932 - 4.40466i) q^{95} +10.1447 q^{97} +0.508536 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 40 q^{9}+O(q^{10}) $ 40 * q + 40 * q^9	$ 40 q + 40 q^{9} + 16 q^{25} - 16 q^{35} + 8 q^{49} + 32 q^{51} - 24 q^{65} - 72 q^{81} + 128 q^{99}+O(q^{100}) $ 40 * q + 40 * q^9 + 16 * q^25 - 16 * q^35 + 8 * q^49 + 32 * q^51 - 24 * q^65 - 72 * q^81 + 128 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.91053			−1.68039			−0.840197	−	0.542281i	$-0.817561\pi$
$3$	−2.91053			−1.68039			−0.840197	+	0.542281i	$0.817561\pi$
$4$		0			0
$5$	−1.83654	−	1.27559i	−0.821324	−	0.570462i
$5$	−1.83654	−	1.27559i	−0.821324	−	0.570462i
$6$		0			0
$7$	−1.52252	+	2.16378i	−0.575459	+	0.817831i
$7$	−1.52252	+	2.16378i	−0.575459	+	0.817831i
$8$		0			0
$9$	5.47117			1.82372
$10$		0			0
$11$	0.0929484			0.0280250			0.0140125	−	0.999902i	$-0.495540\pi$
$11$	0.0929484			0.0280250			0.0140125	+	0.999902i	$0.495540\pi$
$12$		0			0
$13$			4.08979i			1.13430i	0.823613	+	0.567152i	$0.191955\pi$
$13$			4.08979i			1.13430i	−0.823613	+	0.567152i	$0.808045\pi$
$14$		0			0
$15$	5.34529	+	3.71264i	1.38015	+	0.958601i
$16$		0			0
$17$	−4.24008			−1.02837			−0.514185	−	0.857680i	$-0.671905\pi$
$17$	−4.24008			−1.02837			−0.514185	+	0.857680i	$0.671905\pi$
$18$		0			0
$19$			2.39835i			0.550220i	0.961413	+	0.275110i	$0.0887142\pi$
$19$			2.39835i			0.550220i	−0.961413	+	0.275110i	$0.911286\pi$
$20$		0			0
$21$	4.43134	−	6.29773i	0.966998	−	1.37428i
$22$		0			0
$23$	−4.52148			−0.942794			−0.471397	−	0.881921i	$-0.656250\pi$
$23$	−4.52148			−0.942794			−0.471397	+	0.881921i	$0.656250\pi$
$24$		0			0
$25$	1.74573	+	4.68534i	0.349146	+	0.937068i
$26$		0			0
$27$	−7.19241			−1.38418
$28$		0			0
$29$			4.35133i			0.808023i	0.914754	+	0.404011i	$0.132384\pi$
$29$			4.35133i			0.808023i	−0.914754	+	0.404011i	$0.867616\pi$
$30$		0			0
$31$	1.10465			0.198401			0.0992004	−	0.995067i	$-0.468372\pi$
$31$	1.10465			0.198401			0.0992004	+	0.995067i	$0.468372\pi$
$32$		0			0
$33$	−0.270529			−0.0470930
$34$		0			0
$35$	5.55626	−	2.03174i	0.939180	−	0.343427i
$36$		0			0
$37$	8.54612			1.40497			0.702487	−	0.711697i	$-0.252073\pi$
$37$	8.54612			1.40497			0.702487	+	0.711697i	$0.252073\pi$
$38$		0			0
$39$		−	11.9034i		−	1.90608i
$40$		0			0
$41$		−	6.10466i		−	0.953388i	−0.879069	−	0.476694i	$-0.841835\pi$
$41$		−	6.10466i		−	0.953388i	0.879069	−	0.476694i	$-0.158165\pi$
$42$		0			0
$43$		−	4.60415i		−	0.702126i	−0.936352	−	0.351063i	$-0.885820\pi$
$43$		−	4.60415i		−	0.702126i	0.936352	−	0.351063i	$-0.114180\pi$
$44$		0			0
$45$	−10.0480	−	6.97898i	−1.49787	−	1.04036i
$46$		0			0
$47$		−	7.93489i		−	1.15742i	−0.815533	−	0.578711i	$-0.803556\pi$
$47$		−	7.93489i		−	1.15742i	0.815533	−	0.578711i	$-0.196444\pi$
$48$		0			0
$49$	−2.36386	−	6.58879i	−0.337694	−	0.941256i
$50$		0			0
$51$	12.3409			1.72807
$52$		0			0
$53$	−14.3740			−1.97442			−0.987209	−	0.159434i	$-0.949033\pi$
$53$	−14.3740			−1.97442			−0.987209	+	0.159434i	$0.949033\pi$
$54$		0			0
$55$	−0.170703	−	0.118564i	−0.0230176	−	0.0159872i
$56$		0			0
$57$		−	6.98047i		−	0.924586i
$58$		0			0
$59$		−	10.6226i		−	1.38295i	−0.722400	−	0.691476i	$-0.756961\pi$
$59$		−	10.6226i		−	1.38295i	0.722400	−	0.691476i	$-0.243039\pi$
$60$		0			0
$61$	−1.92656			−0.246671			−0.123336	−	0.992365i	$-0.539359\pi$
$61$	−1.92656			−0.246671			−0.123336	+	0.992365i	$0.539359\pi$
$62$		0			0
$63$	−8.32997	+	11.8384i	−1.04948	+	1.49150i
$64$		0			0
$65$	5.21690	−	7.51105i	0.647077	−	0.931631i
$66$		0			0
$67$		−	13.1636i		−	1.60819i	−0.594499	−	0.804097i	$-0.702649\pi$
$67$		−	13.1636i		−	1.60819i	0.594499	−	0.804097i	$-0.297351\pi$
$68$		0			0
$69$	13.1599			1.58426
$70$		0			0
$71$			9.75994i			1.15829i	0.815224	+	0.579146i	$0.196614\pi$
$71$			9.75994i			1.15829i	−0.815224	+	0.579146i	$0.803386\pi$
$72$		0			0
$73$	6.19245			0.724772			0.362386	−	0.932028i	$-0.381962\pi$
$73$	6.19245			0.724772			0.362386	+	0.932028i	$0.381962\pi$
$74$		0			0
$75$	−5.08100	−	13.6368i	−0.586703	−	1.57464i
$76$		0			0
$77$	−0.141516	+	0.201119i	−0.0161272	+	0.0229197i
$78$		0			0
$79$			3.42089i			0.384880i	0.981309	+	0.192440i	$0.0616401\pi$
$79$			3.42089i			0.384880i	−0.981309	+	0.192440i	$0.938360\pi$
$80$		0			0
$81$	4.52019			0.502243
$82$		0			0
$83$	11.7848			1.29355			0.646774	−	0.762682i	$-0.276118\pi$
$83$	11.7848			1.29355			0.646774	+	0.762682i	$0.276118\pi$
$84$		0			0
$85$	7.78705	+	5.40860i	0.844624	+	0.586646i
$86$		0			0
$87$		−	12.6647i		−	1.35780i
$88$		0			0
$89$		−	4.46220i		−	0.472992i	−0.971632	−	0.236496i	$-0.924001\pi$
$89$		−	4.46220i		−	0.472992i	0.971632	−	0.236496i	$-0.0759991\pi$
$90$		0			0
$91$	−8.84939	−	6.22679i	−0.927668	−	0.652745i
$92$		0			0
$93$	−3.21511			−0.333391
$94$		0			0
$95$	3.05932	−	4.40466i	0.313879	−	0.451909i
$96$		0			0
$97$	10.1447			1.03004			0.515021	−	0.857177i	$-0.327784\pi$
$97$	10.1447			1.03004			0.515021	+	0.857177i	$0.327784\pi$
$98$		0			0
$99$	0.508536			0.0511098
$100$		0			0
$101$	−7.27752			−0.724140			−0.362070	−	0.932151i	$-0.617930\pi$
$101$	−7.27752			−0.724140			−0.362070	+	0.932151i	$0.617930\pi$
$102$		0			0
$103$			8.57425i			0.844846i	0.906399	+	0.422423i	$0.138820\pi$
$103$			8.57425i			0.844846i	−0.906399	+	0.422423i	$0.861180\pi$
$104$		0			0
$105$	−16.1717	+	5.91343i	−1.57819	+	0.577092i
$106$		0			0
$107$			7.51221i			0.726233i	0.931744	+	0.363117i	$0.118287\pi$
$107$			7.51221i			0.726233i	−0.931744	+	0.363117i	$0.881713\pi$
$108$		0			0
$109$		−	10.1050i		−	0.967886i	−0.875100	−	0.483943i	$-0.839204\pi$
$109$		−	10.1050i		−	0.967886i	0.875100	−	0.483943i	$-0.160796\pi$
$110$		0			0
$111$	−24.8737			−2.36091
$112$		0			0
$113$		−	2.22783i		−	0.209577i	−0.994495	−	0.104788i	$-0.966583\pi$
$113$		−	2.22783i		−	0.209577i	0.994495	−	0.104788i	$-0.0334165\pi$
$114$		0			0
$115$	8.30386	+	5.76756i	0.774339	+	0.537828i
$116$		0			0
$117$			22.3759i			2.06866i
$118$		0			0
$119$	6.45560	−	9.17458i	0.591784	−	0.841032i
$120$		0			0
$121$	−10.9914			−0.999215
$122$		0			0
$123$			17.7678i			1.60207i
$124$		0			0
$125$	2.77048	−	10.8316i	0.247799	−	0.968811i
$126$		0			0
$127$	7.92857			0.703547			0.351773	−	0.936085i	$-0.385579\pi$
$127$	7.92857			0.703547			0.351773	+	0.936085i	$0.385579\pi$
$128$		0			0
$129$			13.4005i			1.17985i
$130$		0			0
$131$			7.55609i			0.660179i	0.943950	+	0.330089i	$0.107079\pi$
$131$			7.55609i			0.660179i	−0.943950	+	0.330089i	$0.892921\pi$
$132$		0			0
$133$	−5.18950	−	3.65154i	−0.449987	−	0.316629i
$134$		0			0
$135$	13.2091	+	9.17458i	1.13686	+	0.789622i
$136$		0			0
$137$		−	18.2856i		−	1.56225i	−0.624377	−	0.781123i	$-0.714647\pi$
$137$		−	18.2856i		−	1.56225i	0.624377	−	0.781123i	$-0.285353\pi$
$138$		0			0
$139$			1.58667i			0.134579i	0.997733	+	0.0672897i	$0.0214352\pi$
$139$			1.58667i			0.134579i	−0.997733	+	0.0672897i	$0.978565\pi$
$140$		0			0
$141$			23.0947i			1.94493i
$142$		0			0
$143$			0.380139i			0.0317888i
$144$		0			0
$145$	5.55053	−	7.99138i	0.460946	−	0.663648i
$146$		0			0
$147$	6.88007	+	19.1769i	0.567459	+	1.58168i
$148$		0			0
$149$		−	11.2114i		−	0.918476i	−0.888313	−	0.459238i	$-0.848123\pi$
$149$		−	11.2114i		−	0.918476i	0.888313	−	0.459238i	$-0.151877\pi$
$150$		0			0
$151$			2.14351i			0.174436i	0.996189	+	0.0872180i	$0.0277977\pi$
$151$			2.14351i			0.174436i	−0.996189	+	0.0872180i	$0.972202\pi$
$152$		0			0
$153$	−23.1982			−1.87546
$154$		0			0
$155$	−2.02873	−	1.40908i	−0.162951	−	0.113180i
$156$		0			0
$157$		−	6.74329i		−	0.538173i	−0.963116	−	0.269087i	$-0.913278\pi$
$157$		−	6.74329i		−	0.538173i	0.963116	−	0.269087i	$-0.0867218\pi$
$158$		0			0
$159$	41.8358			3.31780
$160$		0			0
$161$	6.88405	−	9.78347i	0.542539	−	0.771046i
$162$		0			0
$163$		−	7.51221i		−	0.588402i	−0.955744	−	0.294201i	$-0.904946\pi$
$163$		−	7.51221i		−	0.588402i	0.955744	−	0.294201i	$-0.0950535\pi$
$164$		0			0
$165$	0.496836	+	0.345084i	0.0386786	+	0.0268648i
$166$		0			0
$167$		−	8.57425i		−	0.663496i	−0.943368	−	0.331748i	$-0.892362\pi$
$167$		−	8.57425i		−	0.663496i	0.943368	−	0.331748i	$-0.107638\pi$
$168$		0			0
$169$	−3.72638			−0.286645
$170$		0			0
$171$			13.1218i			1.00345i
$172$		0			0
$173$			23.9145i			1.81819i	0.416590	+	0.909094i	$0.363225\pi$
$173$			23.9145i			1.81819i	−0.416590	+	0.909094i	$0.636775\pi$
$174$		0			0
$175$	−12.7959	−	3.35616i	−0.967283	−	0.253702i
$176$		0			0
$177$			30.9175i			2.32390i
$178$		0			0
$179$	11.5489			0.863207			0.431603	−	0.902064i	$-0.357948\pi$
$179$	11.5489			0.863207			0.431603	+	0.902064i	$0.357948\pi$
$180$		0			0
$181$	−10.8919			−0.809590			−0.404795	−	0.914407i	$-0.632657\pi$
$181$	−10.8919			−0.809590			−0.404795	+	0.914407i	$0.632657\pi$
$182$		0			0
$183$	5.60732			0.414505
$184$		0			0
$185$	−15.6953	−	10.9014i	−1.15394	−	0.801484i
$186$		0			0
$187$	−0.394108			−0.0288200
$188$		0			0
$189$	10.9506	−	15.5628i	0.796538	−	1.13202i
$190$		0			0
$191$		−	3.42089i		−	0.247527i	−0.992312	−	0.123763i	$-0.960504\pi$
$191$		−	3.42089i		−	0.247527i	0.992312	−	0.123763i	$-0.0394964\pi$
$192$		0			0
$193$			18.1232i			1.30454i	0.757987	+	0.652270i	$0.226183\pi$
$193$			18.1232i			1.30454i	−0.757987	+	0.652270i	$0.773817\pi$
$194$		0			0
$195$	−15.1839	+	21.8611i	−1.08734	+	1.56551i
$196$		0			0
$197$	18.9251			1.34836			0.674179	−	0.738568i	$-0.264497\pi$
$197$	18.9251			1.34836			0.674179	+	0.738568i	$0.264497\pi$
$198$		0			0
$199$	−6.52608			−0.462622			−0.231311	−	0.972880i	$-0.574301\pi$
$199$	−6.52608			−0.462622			−0.231311	+	0.972880i	$0.574301\pi$
$200$		0			0
$201$			38.3131i			2.70240i
$202$		0			0
$203$	−9.41532	−	6.62500i	−0.660826	−	0.464984i
$204$		0			0
$205$	−7.78705	+	11.2114i	−0.543871	+	0.783040i
$206$		0			0
$207$	−24.7378			−1.71939
$208$		0			0
$209$			0.222923i			0.0154199i
$210$		0			0
$211$	4.66349			0.321048			0.160524	−	0.987032i	$-0.448682\pi$
$211$	4.66349			0.321048			0.160524	+	0.987032i	$0.448682\pi$
$212$		0			0
$213$		−	28.4066i		−	1.94639i
$214$		0			0
$215$	−5.87302	+	8.45569i	−0.400536	+	0.576673i
$216$		0			0
$217$	−1.68185	+	2.39021i	−0.114171	+	0.162258i
$218$		0			0
$219$	−18.0233			−1.21790
$220$		0			0
$221$		−	17.3410i		−	1.16648i
$222$		0			0
$223$		−	8.18714i		−	0.548252i	−0.961694	−	0.274126i	$-0.911612\pi$
$223$		−	8.18714i		−	0.548252i	0.961694	−	0.274126i	$-0.0883885\pi$
$224$		0			0
$225$	9.55120	+	25.6343i	0.636746	+	1.70895i
$226$		0			0
$227$	−0.687619			−0.0456389			−0.0228195	−	0.999740i	$-0.507264\pi$
$227$	−0.687619			−0.0456389			−0.0228195	+	0.999740i	$0.507264\pi$
$228$		0			0
$229$	5.06822			0.334918			0.167459	−	0.985879i	$-0.446444\pi$
$229$	5.06822			0.334918			0.167459	+	0.985879i	$0.446444\pi$
$230$		0			0
$231$	0.411886	−	0.585364i	0.0271001	−	0.0385141i
$232$		0			0
$233$		−	15.8954i		−	1.04134i	−0.853757	−	0.520672i	$-0.825682\pi$
$233$		−	15.8954i		−	1.04134i	0.853757	−	0.520672i	$-0.174318\pi$
$234$		0			0
$235$	−10.1217	+	14.5727i	−0.660265	+	0.950619i
$236$		0			0
$237$		−	9.95658i		−	0.646750i
$238$		0			0
$239$			12.2302i			0.791106i	0.918443	+	0.395553i	$0.129447\pi$
$239$			12.2302i			0.791106i	−0.918443	+	0.395553i	$0.870553\pi$
$240$		0			0
$241$			3.80710i			0.245237i	0.992454	+	0.122618i	$0.0391291\pi$
$241$			3.80710i			0.245237i	−0.992454	+	0.122618i	$0.960871\pi$
$242$		0			0
$243$	8.42108			0.540213
$244$		0			0
$245$	−4.06330	+	15.1159i	−0.259594	+	0.965718i
$246$		0			0
$247$	−9.80876			−0.624116
$248$		0			0
$249$	−34.2999			−2.17367
$250$		0			0
$251$		−	11.5411i		−	0.728468i	−0.931307	−	0.364234i	$-0.881331\pi$
$251$		−	11.5411i		−	0.728468i	0.931307	−	0.364234i	$-0.118669\pi$
$252$		0			0
$253$	−0.420264			−0.0264218
$254$		0			0
$255$	−22.6644	−	15.7419i	−1.41930	−	0.985796i
$256$		0			0
$257$	20.9753			1.30840			0.654201	−	0.756321i	$-0.273005\pi$
$257$	20.9753			1.30840			0.654201	+	0.756321i	$0.273005\pi$
$258$		0			0
$259$	−13.0117	+	18.4919i	−0.808505	+	1.14903i
$260$		0			0
$261$			23.8069i			1.47361i
$262$		0			0
$263$	−1.08584			−0.0669558			−0.0334779	−	0.999439i	$-0.510658\pi$
$263$	−1.08584			−0.0669558			−0.0334779	+	0.999439i	$0.510658\pi$
$264$		0			0
$265$	26.3983	+	18.3353i	1.62164	+	1.12633i
$266$		0			0
$267$			12.9874i			0.794813i
$268$		0			0
$269$	6.22378			0.379470			0.189735	−	0.981835i	$-0.439237\pi$
$269$	6.22378			0.379470			0.189735	+	0.981835i	$0.439237\pi$
$270$		0			0
$271$	28.3667			1.72316			0.861578	−	0.507624i	$-0.169476\pi$
$271$	28.3667			1.72316			0.861578	+	0.507624i	$0.169476\pi$
$272$		0			0
$273$	25.7564	+	18.1232i	1.55885	+	1.09687i
$274$		0			0
$275$	0.162263	+	0.435495i	0.00978482	+	0.0262613i
$276$		0			0
$277$	−0.917100			−0.0551032			−0.0275516	−	0.999620i	$-0.508771\pi$
$277$	−0.917100			−0.0551032			−0.0275516	+	0.999620i	$0.508771\pi$
$278$		0			0
$279$	6.04372			0.361828
$280$		0			0
$281$	−11.8342			−0.705969			−0.352984	−	0.935629i	$-0.614833\pi$
$281$	−11.8342			−0.705969			−0.352984	+	0.935629i	$0.614833\pi$
$282$		0			0
$283$	−22.0343			−1.30980			−0.654901	−	0.755715i	$-0.727290\pi$
$283$	−22.0343			−1.30980			−0.654901	+	0.755715i	$0.727290\pi$
$284$		0			0
$285$	−8.90423	+	12.8199i	−0.527441	+	0.759385i
$286$		0			0
$287$	13.2091	+	9.29447i	0.779710	+	0.548635i
$288$		0			0
$289$	0.978238			0.0575434
$290$		0			0
$291$	−29.5265			−1.73088
$292$		0			0
$293$		−	16.2092i		−	0.946953i	−0.880807	−	0.473476i	$-0.842999\pi$
$293$		−	16.2092i		−	0.946953i	0.880807	−	0.473476i	$-0.157001\pi$
$294$		0			0
$295$	−13.5502	+	19.5089i	−0.788921	+	1.13585i
$296$		0			0
$297$	−0.668522			−0.0387916
$298$		0			0
$299$		−	18.4919i		−	1.06941i
$300$		0			0
$301$	9.96235	+	7.00992i	0.574220	+	0.404045i
$302$		0			0
$303$	21.1814			1.21684
$304$		0			0
$305$	3.53821	+	2.45751i	0.202597	+	0.140717i
$306$		0			0
$307$	5.56962			0.317875			0.158938	−	0.987289i	$-0.449193\pi$
$307$	5.56962			0.317875			0.158938	+	0.987289i	$0.449193\pi$
$308$		0			0
$309$		−	24.9556i		−	1.41967i
$310$		0			0
$311$	−8.31354			−0.471418			−0.235709	−	0.971824i	$-0.575741\pi$
$311$	−8.31354			−0.471418			−0.235709	+	0.971824i	$0.575741\pi$
$312$		0			0
$313$	−0.184481			−0.0104275			−0.00521375	−	0.999986i	$-0.501660\pi$
$313$	−0.184481			−0.0104275			−0.00521375	+	0.999986i	$0.501660\pi$
$314$		0			0
$315$	30.3992	−	11.1160i	1.71280	−	0.626315i
$316$		0			0
$317$	−0.135906			−0.00763321			−0.00381661	−	0.999993i	$-0.501215\pi$
$317$	−0.135906			−0.00763321			−0.00381661	+	0.999993i	$0.501215\pi$
$318$		0			0
$319$			0.404449i			0.0226448i
$320$		0			0
$321$		−	21.8645i		−	1.22036i
$322$		0			0
$323$		−	10.1692i		−	0.565829i
$324$		0			0
$325$	−19.1621	+	7.13968i	−1.06292	+	0.396038i
$326$		0			0
$327$			29.4109i			1.62643i
$328$		0			0
$329$	17.1693	+	12.0810i	0.946575	+	0.666049i
$330$		0			0
$331$	−26.0751			−1.43322			−0.716609	−	0.697475i	$-0.754307\pi$
$331$	−26.0751			−1.43322			−0.716609	+	0.697475i	$0.754307\pi$
$332$		0			0
$333$	46.7573			2.56228
$334$		0			0
$335$	−16.7914	+	24.1755i	−0.917413	+	1.32085i
$336$		0			0
$337$		−	3.87932i		−	0.211320i	−0.994402	−	0.105660i	$-0.966305\pi$
$337$		−	3.87932i		−	0.211320i	0.994402	−	0.105660i	$-0.0336955\pi$
$338$		0			0
$339$			6.48416i			0.352171i
$340$		0			0
$341$	0.102675			0.00556018
$342$		0			0
$343$	17.8557	+	4.91671i	0.964117	+	0.265478i
$344$		0			0
$345$	−24.1686	−	16.7866i	−1.30119	−	0.903763i
$346$		0			0
$347$		−	20.4996i		−	1.10047i	−0.835008	−	0.550237i	$-0.814537\pi$
$347$		−	20.4996i		−	1.10047i	0.835008	−	0.550237i	$-0.185463\pi$
$348$		0			0
$349$	14.2823			0.764512			0.382256	−	0.924056i	$-0.375147\pi$
$349$	14.2823			0.764512			0.382256	+	0.924056i	$0.375147\pi$
$350$		0			0
$351$		−	29.4154i		−	1.57008i
$352$		0			0
$353$	17.7313			0.943740			0.471870	−	0.881668i	$-0.343579\pi$
$353$	17.7313			0.943740			0.471870	+	0.881668i	$0.343579\pi$
$354$		0			0
$355$	12.4497	−	17.9245i	0.660761	−	0.951333i
$356$		0			0
$357$	−18.7892	+	26.7029i	−0.994431	+	1.41326i
$358$		0			0
$359$		−	28.2157i		−	1.48917i	−0.667528	−	0.744585i	$-0.732647\pi$
$359$		−	28.2157i		−	1.48917i	0.667528	−	0.744585i	$-0.267353\pi$
$360$		0			0
$361$	13.2479			0.697258
$362$		0			0
$363$	31.9907			1.67907
$364$		0			0
$365$	−11.3727	−	7.89904i	−0.595273	−	0.413455i
$366$		0			0
$367$		−	26.3739i		−	1.37670i	−0.725377	−	0.688352i	$-0.758334\pi$
$367$		−	26.3739i		−	1.37670i	0.725377	−	0.688352i	$-0.241666\pi$
$368$		0			0
$369$		−	33.3996i		−	1.73872i
$370$		0			0
$371$	21.8847	−	31.1021i	1.13620	−	1.61474i
$372$		0			0
$373$	0.213804			0.0110704			0.00553518	−	0.999985i	$-0.498238\pi$
$373$	0.213804			0.0110704			0.00553518	+	0.999985i	$0.498238\pi$
$374$		0			0
$375$	−8.06356	+	31.5258i	−0.416401	+	1.62798i
$376$		0			0
$377$	−17.7960			−0.916543
$378$		0			0
$379$	−0.508536			−0.0261218			−0.0130609	−	0.999915i	$-0.504158\pi$
$379$	−0.508536			−0.0261218			−0.0130609	+	0.999915i	$0.504158\pi$
$380$		0			0
$381$	−23.0763			−1.18224
$382$		0			0
$383$		−	24.8242i		−	1.26846i	−0.773146	−	0.634228i	$-0.781318\pi$
$383$		−	24.8242i		−	1.26846i	0.773146	−	0.634228i	$-0.218682\pi$
$384$		0			0
$385$	0.516445	−	0.188847i	0.0263205	−	0.00962452i
$386$		0			0
$387$		−	25.1901i		−	1.28048i
$388$		0			0
$389$			18.9060i			0.958570i	0.877659	+	0.479285i	$0.159104\pi$
$389$			18.9060i			0.958570i	−0.877659	+	0.479285i	$0.840896\pi$
$390$		0			0
$391$	19.1714			0.969540
$392$		0			0
$393$		−	21.9922i		−	1.10936i
$394$		0			0
$395$	4.36365	−	6.28258i	0.219559	−	0.316111i
$396$		0			0
$397$		−	17.2079i		−	0.863640i	−0.901960	−	0.431820i	$-0.857872\pi$
$397$		−	17.2079i		−	0.863640i	0.901960	−	0.431820i	$-0.142128\pi$
$398$		0			0
$399$	15.1042	+	10.6279i	0.756155	+	0.532061i
$400$		0			0
$401$	11.6687			0.582708			0.291354	−	0.956615i	$-0.405894\pi$
$401$	11.6687			0.582708			0.291354	+	0.956615i	$0.405894\pi$
$402$		0			0
$403$			4.51778i			0.225047i
$404$		0			0
$405$	−8.30149	−	5.76592i	−0.412505	−	0.286511i
$406$		0			0
$407$	0.794348			0.0393744
$408$		0			0
$409$		−	12.8644i		−	0.636105i	−0.948073	−	0.318052i	$-0.896971\pi$
$409$		−	12.8644i		−	0.636105i	0.948073	−	0.318052i	$-0.103029\pi$
$410$		0			0
$411$			53.2208i			2.62519i
$412$		0			0
$413$	22.9850	+	16.1732i	1.13102	+	0.795832i
$414$		0			0
$415$	−21.6432	−	15.0326i	−1.06242	−	0.737920i
$416$		0			0
$417$		−	4.61804i		−	0.226147i
$418$		0			0
$419$			4.44166i			0.216989i	0.994097	+	0.108495i	$0.0346030\pi$
$419$			4.44166i			0.216989i	−0.994097	+	0.108495i	$0.965397\pi$
$420$		0			0
$421$		−	30.4133i		−	1.48226i	−0.671364	−	0.741128i	$-0.734291\pi$
$421$		−	30.4133i		−	1.48226i	0.671364	−	0.741128i	$-0.265709\pi$
$422$		0			0
$423$		−	43.4131i		−	2.11082i
$424$		0			0
$425$	−7.40203	−	19.8662i	−0.359051	−	0.963652i
$426$		0			0
$427$	2.93324	−	4.16866i	0.141949	−	0.201735i
$428$		0			0
$429$		−	1.10641i		−	0.0534178i
$430$		0			0
$431$			18.1648i			0.874968i	0.899226	+	0.437484i	$0.144130\pi$
$431$			18.1648i			0.874968i	−0.899226	+	0.437484i	$0.855870\pi$
$432$		0			0
$433$	−17.4153			−0.836926			−0.418463	−	0.908234i	$-0.637431\pi$
$433$	−17.4153			−0.836926			−0.418463	+	0.908234i	$0.637431\pi$
$434$		0			0
$435$	−16.1550	+	23.2591i	−0.774571	+	1.11519i
$436$		0			0
$437$		−	10.8441i		−	0.518744i
$438$		0			0
$439$	−34.2855			−1.63636			−0.818180	−	0.574962i	$-0.805017\pi$
$439$	−34.2855			−1.63636			−0.818180	+	0.574962i	$0.805017\pi$
$440$		0			0
$441$	−12.9331	−	36.0484i	−0.615860	−	1.71659i
$442$		0			0
$443$			12.2027i			0.579769i	0.957062	+	0.289885i	$0.0936170\pi$
$443$			12.2027i			0.579769i	−0.957062	+	0.289885i	$0.906383\pi$
$444$		0			0
$445$	−5.69194	+	8.19499i	−0.269824	+	0.388480i
$446$		0			0
$447$			32.6312i			1.54340i
$448$		0			0
$449$	12.3919			0.584809			0.292405	−	0.956295i	$-0.405545\pi$
$449$	12.3919			0.584809			0.292405	+	0.956295i	$0.405545\pi$
$450$		0			0
$451$		−	0.567418i		−	0.0267187i
$452$		0			0
$453$		−	6.23873i		−	0.293121i
$454$		0			0
$455$	8.30939	+	22.7239i	0.389550	+	1.06531i
$456$		0			0
$457$			10.6974i			0.500404i	0.968194	+	0.250202i	$0.0804970\pi$
$457$			10.6974i			0.500404i	−0.968194	+	0.250202i	$0.919503\pi$
$458$		0			0
$459$	30.4964			1.42345
$460$		0			0
$461$	−15.0677			−0.701775			−0.350887	−	0.936418i	$-0.614120\pi$
$461$	−15.0677			−0.701775			−0.350887	+	0.936418i	$0.614120\pi$
$462$		0			0
$463$	−19.4135			−0.902222			−0.451111	−	0.892468i	$-0.648972\pi$
$463$	−19.4135			−0.902222			−0.451111	+	0.892468i	$0.648972\pi$
$464$		0			0
$465$	5.90467	+	4.10117i	0.273822	+	0.190187i
$466$		0			0
$467$	−24.7528			−1.14542			−0.572712	−	0.819756i	$-0.694109\pi$
$467$	−24.7528			−1.14542			−0.572712	+	0.819756i	$0.694109\pi$
$468$		0			0
$469$	28.4832	+	20.0419i	1.31523	+	0.925449i
$470$		0			0
$471$			19.6265i			0.904343i
$472$		0			0
$473$		−	0.427948i		−	0.0196771i
$474$		0			0
$475$	−11.2371	+	4.18688i	−0.515593	+	0.192107i
$476$		0			0
$477$	−78.6425			−3.60079
$478$		0			0
$479$	4.13772			0.189057			0.0945285	−	0.995522i	$-0.469866\pi$
$479$	4.13772			0.189057			0.0945285	+	0.995522i	$0.469866\pi$
$480$		0			0
$481$			34.9518i			1.59367i
$482$		0			0
$483$	−20.0362	+	28.4751i	−0.911679	+	1.29566i
$484$		0			0
$485$	−18.6312	−	12.9405i	−0.845999	−	0.587600i
$486$		0			0
$487$	28.8687			1.30816			0.654082	−	0.756424i	$-0.273055\pi$
$487$	28.8687			1.30816			0.654082	+	0.756424i	$0.273055\pi$
$488$		0			0
$489$			21.8645i			0.988747i
$490$		0			0
$491$	31.0586			1.40165			0.700827	−	0.713331i	$-0.252814\pi$
$491$	31.0586			1.40165			0.700827	+	0.713331i	$0.252814\pi$
$492$		0			0
$493$		−	18.4500i		−	0.830946i
$494$		0			0
$495$	−0.933945	−	0.648685i	−0.0419777	−	0.0291562i
$496$		0			0
$497$	−21.1183	−	14.8597i	−0.947286	−	0.666549i
$498$		0			0
$499$	−16.9068			−0.756854			−0.378427	−	0.925631i	$-0.623535\pi$
$499$	−16.9068			−0.756854			−0.378427	+	0.925631i	$0.623535\pi$
$500$		0			0
$501$			24.9556i			1.11493i
$502$		0			0
$503$		−	8.24998i		−	0.367848i	−0.982940	−	0.183924i	$-0.941120\pi$
$503$		−	8.24998i		−	0.367848i	0.982940	−	0.183924i	$-0.0588801\pi$
$504$		0			0
$505$	13.3654	+	9.28315i	0.594754	+	0.413095i
$506$		0			0
$507$	10.8457			0.481676
$508$		0			0
$509$	24.1632			1.07102			0.535508	−	0.844530i	$-0.320120\pi$
$509$	24.1632			1.07102			0.535508	+	0.844530i	$0.320120\pi$
$510$		0			0
$511$	−9.42814	+	13.3991i	−0.417077	+	0.592741i
$512$		0			0
$513$		−	17.2499i		−	0.761603i
$514$		0			0
$515$	10.9372	−	15.7469i	0.481953	−	0.693893i
$516$		0			0
$517$		−	0.737535i		−	0.0324367i
$518$		0			0
$519$		−	69.6039i		−	3.05527i
$520$		0			0
$521$			36.8884i			1.61611i	0.589108	+	0.808055i	$0.299479\pi$
$521$			36.8884i			1.61611i	−0.589108	+	0.808055i	$0.700521\pi$
$522$		0			0
$523$	6.36172			0.278179			0.139089	−	0.990280i	$-0.455582\pi$
$523$	6.36172			0.278179			0.139089	+	0.990280i	$0.455582\pi$
$524$		0			0
$525$	37.2429	+	9.76819i	1.62542	+	0.426319i
$526$		0			0
$527$	−4.68379			−0.204029
$528$		0			0
$529$	−2.55622			−0.111140
$530$		0			0
$531$		−	58.1183i		−	2.52212i
$532$		0			0
$533$	24.9668			1.08143
$534$		0			0
$535$	9.58252	−	13.7965i	0.414288	−	0.596473i
$536$		0			0
$537$	−33.6134			−1.45053
$538$		0			0
$539$	−0.219717	−	0.612417i	−0.00946387	−	0.0263787i
$540$		0			0
$541$		−	31.6568i		−	1.36103i	−0.732733	−	0.680516i	$-0.761756\pi$
$541$		−	31.6568i		−	1.36103i	0.732733	−	0.680516i	$-0.238244\pi$
$542$		0			0
$543$	31.7012			1.36043
$544$		0			0
$545$	−12.8899	+	18.5582i	−0.552142	+	0.794948i
$546$		0			0
$547$			40.1397i			1.71625i	0.513441	+	0.858125i	$0.328371\pi$
$547$			40.1397i			1.71625i	−0.513441	+	0.858125i	$0.671629\pi$
$548$		0			0
$549$	−10.5406			−0.449860
$550$		0			0
$551$	−10.4360			−0.444590
$552$		0			0
$553$	−7.40203	−	5.20837i	−0.314767	−	0.221483i
$554$		0			0
$555$	45.6815	+	31.7287i	1.93907	+	1.34681i
$556$		0			0
$557$	−0.601042			−0.0254670			−0.0127335	−	0.999919i	$-0.504053\pi$
$557$	−0.601042			−0.0254670			−0.0127335	+	0.999919i	$0.504053\pi$
$558$		0			0
$559$	18.8300			0.796424
$560$		0			0
$561$	1.14706			0.0484290
$562$		0			0
$563$	−25.0288			−1.05484			−0.527419	−	0.849605i	$-0.676840\pi$
$563$	−25.0288			−1.05484			−0.527419	+	0.849605i	$0.676840\pi$
$564$		0			0
$565$	−2.84180	+	4.09149i	−0.119556	+	0.172130i
$566$		0			0
$567$	−6.88209	+	9.78068i	−0.289020	+	0.410750i
$568$		0			0
$569$	14.2638			0.597969			0.298984	−	0.954258i	$-0.403352\pi$
$569$	14.2638			0.597969			0.298984	+	0.954258i	$0.403352\pi$
$570$		0			0
$571$	25.2375			1.05616			0.528079	−	0.849196i	$-0.322913\pi$
$571$	25.2375			1.05616			0.528079	+	0.849196i	$0.322913\pi$
$572$		0			0
$573$			9.95658i			0.415942i
$574$		0			0
$575$	−7.89329	−	21.1847i	−0.329173	−	0.883462i
$576$		0			0
$577$	−8.46289			−0.352315			−0.176157	−	0.984362i	$-0.556367\pi$
$577$	−8.46289			−0.352315			−0.176157	+	0.984362i	$0.556367\pi$
$578$		0			0
$579$		−	52.7482i		−	2.19214i
$580$		0			0
$581$	−17.9426	+	25.4996i	−0.744384	+	1.05790i
$582$		0			0
$583$	−1.33604			−0.0553330
$584$		0			0
$585$	28.5426	−	41.0942i	1.18009	−	1.69904i
$586$		0			0
$587$	−12.5948			−0.519843			−0.259922	−	0.965630i	$-0.583697\pi$
$587$	−12.5948			−0.519843			−0.259922	+	0.965630i	$0.583697\pi$
$588$		0			0
$589$			2.64934i			0.109164i
$590$		0			0
$591$	−55.0821			−2.26577
$592$		0			0
$593$	−23.8425			−0.979096			−0.489548	−	0.871976i	$-0.662838\pi$
$593$	−23.8425			−0.979096			−0.489548	+	0.871976i	$0.662838\pi$
$594$		0			0
$595$	−23.5590	+	8.61473i	−0.965823	+	0.353169i
$596$		0			0
$597$	18.9943			0.777387
$598$		0			0
$599$		−	30.8246i		−	1.25946i	−0.776814	−	0.629730i	$-0.783166\pi$
$599$		−	30.8246i		−	1.25946i	0.776814	−	0.629730i	$-0.216834\pi$
$600$		0			0
$601$		−	30.7889i		−	1.25591i	−0.778251	−	0.627953i	$-0.783893\pi$
$601$		−	30.7889i		−	1.25591i	0.778251	−	0.627953i	$-0.216107\pi$
$602$		0			0
$603$		−	72.0204i		−	2.93290i
$604$		0			0
$605$	20.1860	+	14.0205i	0.820679	+	0.570014i
$606$		0			0
$607$			25.4974i			1.03491i	0.855712	+	0.517453i	$0.173120\pi$
$607$			25.4974i			1.03491i	−0.855712	+	0.517453i	$0.826880\pi$
$608$		0			0
$609$	27.4035	+	19.2822i	1.11045	+	0.781356i
$610$		0			0
$611$	32.4520			1.31287
$612$		0			0
$613$	−26.9818			−1.08978			−0.544891	−	0.838507i	$-0.683429\pi$
$613$	−26.9818			−1.08978			−0.544891	+	0.838507i	$0.683429\pi$
$614$		0			0
$615$	22.6644	−	32.6312i	0.913918	−	1.31582i
$616$		0			0
$617$		−	1.03337i		−	0.0416019i	−0.999784	−	0.0208010i	$-0.993378\pi$
$617$		−	1.03337i		−	0.0416019i	0.999784	−	0.0208010i	$-0.00662163\pi$
$618$		0			0
$619$			6.16501i			0.247793i	0.992295	+	0.123896i	$0.0395391\pi$
$619$			6.16501i			0.247793i	−0.992295	+	0.123896i	$0.960461\pi$
$620$		0			0
$621$	32.5203			1.30500
$622$		0			0
$623$	9.65520	+	6.79379i	0.386828	+	0.272188i
$624$		0			0
$625$	−18.9048	+	16.3587i	−0.756194	+	0.654348i
$626$		0			0
$627$		−	0.648823i		−	0.0259115i
$628$		0			0
$629$	−36.2362			−1.44483
$630$		0			0
$631$			1.30635i			0.0520048i	0.999662	+	0.0260024i	$0.00827776\pi$
$631$			1.30635i			0.0520048i	−0.999662	+	0.0260024i	$0.991722\pi$
$632$		0			0
$633$	−13.5732			−0.539487
$634$		0			0
$635$	−14.5611	−	10.1136i	−0.577840	−	0.401347i
$636$		0			0
$637$	26.9468	−	9.66768i	1.06767	−	0.383048i
$638$		0			0
$639$			53.3983i			2.11240i
$640$		0			0
$641$	24.5606			0.970084			0.485042	−	0.874491i	$-0.338804\pi$
$641$	24.5606			0.970084			0.485042	+	0.874491i	$0.338804\pi$
$642$		0			0
$643$	−13.0725			−0.515530			−0.257765	−	0.966208i	$-0.582986\pi$
$643$	−13.0725			−0.515530			−0.257765	+	0.966208i	$0.582986\pi$
$644$		0			0
$645$	17.0936	−	24.6105i	0.673059	−	0.969038i
$646$		0			0
$647$			13.2349i			0.520318i	0.965566	+	0.260159i	$0.0837750\pi$
$647$			13.2349i			0.520318i	−0.965566	+	0.260159i	$0.916225\pi$
$648$		0			0
$649$		−	0.987358i		−	0.0387572i
$650$		0			0
$651$	4.89507	−	6.95678i	0.191853	−	0.272658i
$652$		0			0
$653$	−14.5528			−0.569495			−0.284748	−	0.958603i	$-0.591910\pi$
$653$	−14.5528			−0.569495			−0.284748	+	0.958603i	$0.591910\pi$
$654$		0			0
$655$	9.63848	−	13.8770i	0.376607	−	0.542221i
$656$		0			0
$657$	33.8800			1.32178
$658$		0			0
$659$	7.39396			0.288028			0.144014	−	0.989576i	$-0.453999\pi$
$659$	7.39396			0.288028			0.144014	+	0.989576i	$0.453999\pi$
$660$		0			0
$661$	2.37065			0.0922074			0.0461037	−	0.998937i	$-0.485320\pi$
$661$	2.37065			0.0922074			0.0461037	+	0.998937i	$0.485320\pi$
$662$		0			0
$663$			50.4715i			1.96015i
$664$		0			0
$665$	4.87283	+	13.3259i	0.188960	+	0.516755i
$666$		0			0
$667$		−	19.6745i		−	0.761799i
$668$		0			0
$669$			23.8289i			0.921279i
$670$		0			0
$671$	−0.179071			−0.00691296
$672$		0			0
$673$		−	11.4258i		−	0.440431i	−0.975451	−	0.220216i	$-0.929324\pi$
$673$		−	11.4258i		−	0.440431i	0.975451	−	0.220216i	$-0.0706761\pi$
$674$		0			0
$675$	−12.5560	−	33.6989i	−0.483281	−	1.29707i
$676$		0			0
$677$			5.36512i			0.206198i	0.994671	+	0.103099i	$0.0328759\pi$
$677$			5.36512i			0.206198i	−0.994671	+	0.103099i	$0.967124\pi$
$678$		0			0
$679$	−15.4456	+	21.9510i	−0.592747	+	0.842400i
$680$		0			0
$681$	2.00134			0.0766913
$682$		0			0
$683$			7.24957i			0.277397i	0.990335	+	0.138699i	$0.0442919\pi$
$683$			7.24957i			0.277397i	−0.990335	+	0.138699i	$0.955708\pi$
$684$		0			0
$685$	−23.3250	+	33.5822i	−0.891202	+	1.28311i
$686$		0			0
$687$	−14.7512			−0.562794
$688$		0			0
$689$		−	58.7865i		−	2.23959i
$690$		0			0
$691$			48.8381i			1.85789i	0.370220	+	0.928944i	$0.379282\pi$
$691$			48.8381i			1.85789i	−0.370220	+	0.928944i	$0.620718\pi$
$692$		0			0
$693$	−0.774257	+	1.10036i	−0.0294116	+	0.0417992i
$694$		0			0
$695$	2.02394	−	2.91398i	0.0767725	−	0.110533i
$696$		0			0
$697$			25.8842i			0.980435i
$698$		0			0
$699$			46.2640i			1.74987i
$700$		0			0
$701$			5.22242i			0.197248i	0.995125	+	0.0986240i	$0.0314441\pi$
$701$			5.22242i			0.197248i	−0.995125	+	0.0986240i	$0.968556\pi$
$702$		0			0
$703$			20.4966i			0.773044i
$704$		0			0
$705$	29.4594	−	42.4143i	1.10951	−	1.59741i
$706$		0			0
$707$	11.0802	−	15.7469i	0.416713	−	0.592224i
$708$		0			0
$709$		−	31.5014i		−	1.18306i	−0.806283	−	0.591530i	$-0.798524\pi$
$709$		−	31.5014i		−	1.18306i	0.806283	−	0.591530i	$-0.201476\pi$
$710$		0			0
$711$			18.7163i			0.701914i
$712$		0			0
$713$	−4.99465			−0.187051
$714$		0			0
$715$	0.484902	−	0.698140i	0.0181343	−	0.0261089i
$716$		0			0
$717$		−	35.5964i		−	1.32937i
$718$		0			0
$719$	7.20252			0.268609			0.134304	−	0.990940i	$-0.457120\pi$
$719$	7.20252			0.268609			0.134304	+	0.990940i	$0.457120\pi$
$720$		0			0
$721$	−18.5528	−	13.0545i	−0.690941	−	0.486174i
$722$		0			0
$723$		−	11.0807i		−	0.412094i
$724$		0			0
$725$	−20.3875	+	7.59626i	−0.757172	+	0.282118i
$726$		0			0
$727$		−	20.0473i		−	0.743515i	−0.928330	−	0.371757i	$-0.878755\pi$
$727$		−	20.0473i		−	0.743515i	0.928330	−	0.371757i	$-0.121245\pi$
$728$		0			0
$729$	−38.0704			−1.41001
$730$		0			0
$731$			19.5219i			0.722045i
$732$		0			0
$733$		−	9.51770i		−	0.351544i	−0.984431	−	0.175772i	$-0.943758\pi$
$733$		−	9.51770i		−	0.351544i	0.984431	−	0.175772i	$-0.0562422\pi$
$734$		0			0
$735$	11.8263	−	43.9952i	0.436221	−	1.62279i
$736$		0			0
$737$		−	1.22354i		−	0.0450696i
$738$		0			0
$739$	36.0009			1.32432			0.662158	−	0.749364i	$-0.269641\pi$
$739$	36.0009			1.32432			0.662158	+	0.749364i	$0.269641\pi$
$740$		0			0
$741$	28.5487			1.04876
$742$		0			0
$743$	−30.4574			−1.11737			−0.558686	−	0.829379i	$-0.688694\pi$
$743$	−30.4574			−1.11737			−0.558686	+	0.829379i	$0.688694\pi$
$744$		0			0
$745$	−14.3012	+	20.5902i	−0.523956	+	0.754366i
$746$		0			0
$747$	64.4766			2.35907
$748$		0			0
$749$	−16.2548	−	11.4375i	−0.593936	−	0.417917i
$750$		0			0
$751$		−	36.5140i		−	1.33241i	−0.745767	−	0.666207i	$-0.767917\pi$
$751$		−	36.5140i		−	1.33241i	0.745767	−	0.666207i	$-0.232083\pi$
$752$		0			0
$753$			33.5907i			1.22411i
$754$		0			0
$755$	2.73424	−	3.93663i	0.0995091	−	0.143269i
$756$		0			0
$757$	37.9558			1.37953			0.689763	−	0.724035i	$-0.257715\pi$
$757$	37.9558			1.37953			0.689763	+	0.724035i	$0.257715\pi$
$758$		0			0
$759$	1.22319			0.0443990
$760$		0			0
$761$		−	8.65361i		−	0.313693i	−0.987623	−	0.156846i	$-0.949867\pi$
$761$		−	8.65361i		−	0.313693i	0.987623	−	0.156846i	$-0.0501328\pi$
$762$		0			0
$763$	21.8650	+	15.3851i	0.791567	+	0.556978i
$764$		0			0
$765$	42.6043	+	29.5914i	1.54036	+	1.06988i
$766$		0			0
$767$	43.4444			1.56869
$768$		0			0
$769$			19.0593i			0.687297i	0.939098	+	0.343648i	$0.111663\pi$
$769$			19.0593i			0.687297i	−0.939098	+	0.343648i	$0.888337\pi$
$770$		0			0
$771$	−61.0491			−2.19863
$772$		0			0
$773$			31.6953i			1.14000i	0.821645	+	0.570000i	$0.193057\pi$
$773$			31.6953i			1.14000i	−0.821645	+	0.570000i	$0.806943\pi$
$774$		0			0
$775$	1.92842	+	5.17565i	0.0692709	+	0.185915i
$776$		0			0
$777$	37.8708	−	53.8212i	1.35861	−	1.93082i
$778$		0			0
$779$	14.6411			0.524573
$780$		0			0
$781$			0.907170i			0.0324611i
$782$		0			0
$783$		−	31.2966i		−	1.11845i
$784$		0			0
$785$	−8.60168	+	12.3843i	−0.307007	+	0.442015i
$786$		0			0
$787$	−1.34681			−0.0480085			−0.0240043	−	0.999712i	$-0.507642\pi$
$787$	−1.34681			−0.0480085			−0.0240043	+	0.999712i	$0.507642\pi$
$788$		0			0
$789$	3.16037			0.112512
$790$		0			0
$791$	4.82053	+	3.39192i	0.171398	+	0.120603i
$792$		0			0
$793$		−	7.87925i		−	0.279800i
$794$		0			0
$795$	−76.8330	−	53.3655i	−2.72499	−	1.89268i
$796$		0			0
$797$		−	44.1264i		−	1.56304i	−0.623882	−	0.781519i	$-0.714445\pi$
$797$		−	44.1264i		−	1.56304i	0.623882	−	0.781519i	$-0.285555\pi$
$798$		0			0
$799$			33.6445i			1.19026i
$800$		0			0
$801$		−	24.4135i		−	0.862607i
$802$		0			0
$803$	0.575578			0.0203117
$804$		0			0
$805$	−25.1225	+	9.18647i	−0.885453	+	0.323780i
$806$		0			0
$807$	−18.1145			−0.637660
$808$		0			0
$809$	−22.6554			−0.796521			−0.398261	−	0.917272i	$-0.630386\pi$
$809$	−22.6554			−0.796521			−0.398261	+	0.917272i	$0.630386\pi$
$810$		0			0
$811$		−	23.8620i		−	0.837909i	−0.908007	−	0.418954i	$-0.862397\pi$
$811$		−	23.8620i		−	0.837909i	0.908007	−	0.418954i	$-0.137603\pi$
$812$		0			0
$813$	−82.5622			−2.89558
$814$		0			0
$815$	−9.58252	+	13.7965i	−0.335661	+	0.483269i
$816$		0			0
$817$	11.0424			0.386324
$818$		0			0
$819$	−48.4165	−	34.0678i	−1.69181	−	1.19043i
$820$		0			0
$821$		−	26.3396i		−	0.919260i	−0.888111	−	0.459630i	$-0.847982\pi$
$821$		−	26.3396i		−	0.919260i	0.888111	−	0.459630i	$-0.152018\pi$
$822$		0			0
$823$	−16.2057			−0.564896			−0.282448	−	0.959283i	$-0.591147\pi$
$823$	−16.2057			−0.564896			−0.282448	+	0.959283i	$0.591147\pi$
$824$		0			0
$825$	−0.472271	−	1.26752i	−0.0164424	−	0.0441294i
$826$		0			0
$827$		−	10.3304i		−	0.359222i	−0.983738	−	0.179611i	$-0.942516\pi$
$827$		−	10.3304i		−	0.359222i	0.983738	−	0.179611i	$-0.0574840\pi$
$828$		0			0
$829$	−48.5037			−1.68460			−0.842302	−	0.539007i	$-0.818800\pi$
$829$	−48.5037			−1.68460			−0.842302	+	0.539007i	$0.818800\pi$
$830$		0			0
$831$	2.66924			0.0925951
$832$		0			0
$833$	10.0229	+	27.9370i	0.347274	+	0.967959i
$834$		0			0
$835$	−10.9372	+	15.7469i	−0.378499	+	0.544945i
$836$		0			0
$837$	−7.94508			−0.274622
$838$		0			0
$839$	−25.6975			−0.887176			−0.443588	−	0.896231i	$-0.646295\pi$
$839$	−25.6975			−0.887176			−0.443588	+	0.896231i	$0.646295\pi$
$840$		0			0
$841$	10.0659			0.347100
$842$		0			0
$843$	34.4437			1.18631
$844$		0			0
$845$	6.84363	+	4.75334i	0.235428	+	0.163520i
$846$		0			0
$847$	16.7346	−	23.7828i	0.575007	−	0.817188i
$848$		0			0
$849$	64.1314			2.20098
$850$		0			0
$851$	−38.6411			−1.32460
$852$		0			0
$853$			36.3693i			1.24526i	0.782516	+	0.622630i	$0.213936\pi$
$853$			36.3693i			1.24526i	−0.782516	+	0.622630i	$0.786064\pi$
$854$		0			0
$855$	16.7380	−	24.0986i	0.572429	−	0.824156i
$856$		0			0
$857$	−7.87431			−0.268981			−0.134491	−	0.990915i	$-0.542940\pi$
$857$	−7.87431			−0.268981			−0.134491	+	0.990915i	$0.542940\pi$
$858$		0			0
$859$			6.61172i			0.225589i	0.993618	+	0.112795i	$0.0359802\pi$
$859$			6.61172i			0.225589i	−0.993618	+	0.112795i	$0.964020\pi$
$860$		0			0
$861$	−38.4455	−	27.0518i	−1.31022	−	0.921924i
$862$		0			0
$863$	−52.1673			−1.77580			−0.887898	−	0.460040i	$-0.847835\pi$
$863$	−52.1673			−1.77580			−0.887898	+	0.460040i	$0.847835\pi$
$864$		0			0
$865$	30.5052	−	43.9199i	1.03721	−	1.49332i
$866$		0			0
$867$	−2.84719			−0.0966956
$868$		0			0
$869$			0.317966i			0.0107863i
$870$		0			0
$871$	53.8365			1.82418
$872$		0			0
$873$	55.5036			1.87851
$874$		0			0
$875$	19.2191	+	22.4861i	0.649725	+	0.760169i
$876$		0			0
$877$	−28.4925			−0.962125			−0.481063	−	0.876686i	$-0.659749\pi$
$877$	−28.4925			−0.962125			−0.481063	+	0.876686i	$0.659749\pi$
$878$		0			0
$879$			47.1774i			1.59125i
$880$		0			0
$881$		−	33.0865i		−	1.11471i	−0.830274	−	0.557356i	$-0.811816\pi$
$881$		−	33.0865i		−	1.11471i	0.830274	−	0.557356i	$-0.188184\pi$
$882$		0			0
$883$		−	13.6397i		−	0.459013i	−0.973307	−	0.229506i	$-0.926289\pi$
$883$		−	13.6397i		−	0.459013i	0.973307	−	0.229506i	$-0.0737112\pi$
$884$		0			0
$885$	39.4381	−	56.7811i	1.32570	−	1.90868i
$886$		0			0
$887$		−	14.1834i		−	0.476233i	−0.971237	−	0.238116i	$-0.923470\pi$
$887$		−	14.1834i		−	0.476233i	0.971237	−	0.238116i	$-0.0765300\pi$
$888$		0			0
$889$	−12.0714	+	17.1557i	−0.404862	+	0.575382i
$890$		0			0
$891$	0.420144			0.0140754
$892$		0			0
$893$	19.0307			0.636837
$894$		0			0
$895$	−21.2100	−	14.7317i	−0.708972	−	0.492427i
$896$		0			0
$897$			53.8212i			1.79704i
$898$		0			0
$899$			4.80669i			0.160312i
$900$		0			0
$901$	60.9467			2.03043
$902$		0			0
$903$	−28.9957	−	20.4026i	−0.964917	−	0.678955i
$904$		0			0
$905$	20.0034	+	13.8936i	0.664936	+	0.461840i
$906$		0			0
$907$			43.9936i			1.46078i	0.683029	+	0.730391i	$0.260662\pi$
$907$			43.9936i			1.46078i	−0.683029	+	0.730391i	$0.739338\pi$
$908$		0			0
$909$	−39.8166			−1.32063
$910$		0			0
$911$			16.0145i			0.530584i	0.964168	+	0.265292i	$0.0854684\pi$
$911$			16.0145i			0.530584i	−0.964168	+	0.265292i	$0.914532\pi$
$912$		0			0
$913$	1.09538			0.0362517
$914$		0			0
$915$	−10.2980	−	7.15265i	−0.340443	−	0.236459i
$916$		0			0
$917$	−16.3497	−	11.5043i	−0.539914	−	0.379906i
$918$		0			0
$919$		−	8.04985i		−	0.265540i	−0.991147	−	0.132770i	$-0.957613\pi$
$919$		−	8.04985i		−	0.265540i	0.991147	−	0.132770i	$-0.0423872\pi$
$920$		0			0
$921$	−16.2105			−0.534155
$922$		0			0
$923$	−39.9161			−1.31385
$924$		0			0
$925$	14.9192	+	40.0415i	0.490542	+	1.31656i
$926$		0			0
$927$			46.9112i			1.54077i
$928$		0			0
$929$			24.7127i			0.810798i	0.914140	+	0.405399i	$0.132867\pi$
$929$			24.7127i			0.810798i	−0.914140	+	0.405399i	$0.867133\pi$
$930$		0			0
$931$	15.8022	−	5.66936i	0.517898	−	0.185806i
$932$		0			0
$933$	24.1968			0.792168
$934$		0			0
$935$	0.723794	+	0.502721i	0.0236706	+	0.0164407i
$936$		0			0
$937$	−11.4286			−0.373356			−0.186678	−	0.982421i	$-0.559772\pi$
$937$	−11.4286			−0.373356			−0.186678	+	0.982421i	$0.559772\pi$
$938$		0			0
$939$	0.536938			0.0175223
$940$		0			0
$941$	46.2213			1.50677			0.753385	−	0.657580i	$-0.228420\pi$
$941$	46.2213			1.50677			0.753385	+	0.657580i	$0.228420\pi$
$942$		0			0
$943$			27.6021i			0.898848i
$944$		0			0
$945$	−39.9629	+	14.6131i	−1.29999	+	0.475364i
$946$		0			0
$947$		−	29.3217i		−	0.952827i	−0.879221	−	0.476413i	$-0.841937\pi$
$947$		−	29.3217i		−	0.952827i	0.879221	−	0.476413i	$-0.158063\pi$
$948$		0			0
$949$			25.3258i			0.822111i
$950$		0			0
$951$	0.395557			0.0128268
$952$		0			0
$953$		−	53.3364i		−	1.72773i	−0.503719	−	0.863867i	$-0.668035\pi$
$953$		−	53.3364i		−	1.72773i	0.503719	−	0.863867i	$-0.331965\pi$
$954$		0			0
$955$	−4.36365	+	6.28258i	−0.141205	+	0.203300i
$956$		0			0
$957$		−	1.17716i		−	0.0380522i
$958$		0			0
$959$	39.5660	+	27.8403i	1.27765	+	0.899009i
$960$		0			0
$961$	−29.7798			−0.960637
$962$		0			0
$963$			41.1006i			1.32445i
$964$		0			0
$965$	23.1179	−	33.2840i	0.744190	−	1.07145i
$966$		0			0
$967$	49.6379			1.59625			0.798123	−	0.602494i	$-0.205826\pi$
$967$	49.6379			1.59625			0.798123	+	0.602494i	$0.205826\pi$
$968$		0			0
$969$			29.5977i			0.950816i
$970$		0			0
$971$			9.69938i			0.311268i	0.987815	+	0.155634i	$0.0497421\pi$
$971$			9.69938i			0.311268i	−0.987815	+	0.155634i	$0.950258\pi$
$972$		0			0
$973$	−3.43320	−	2.41574i	−0.110063	−	0.0774450i
$974$		0			0
$975$	55.7717	−	20.7802i	1.78612	−	0.665500i
$976$		0			0
$977$			40.7439i			1.30351i	0.758428	+	0.651757i	$0.225968\pi$
$977$			40.7439i			1.30351i	−0.758428	+	0.651757i	$0.774032\pi$
$978$		0			0
$979$		−	0.414754i		−	0.0132556i
$980$		0			0
$981$		−	55.2863i		−	1.76516i
$982$		0			0
$983$			1.23527i			0.0393990i	0.999806	+	0.0196995i	$0.00627095\pi$
$983$			1.23527i			0.0393990i	−0.999806	+	0.0196995i	$0.993729\pi$
$984$		0			0
$985$	−34.7567	−	24.1407i	−1.10744	−	0.769187i
$986$		0			0
$987$	−49.9718	−	35.1622i	−1.59062	−	1.11922i
$988$		0			0
$989$			20.8176i			0.661960i
$990$		0			0
$991$		−	49.1076i		−	1.55995i	−0.625808	−	0.779977i	$-0.715231\pi$
$991$		−	49.1076i		−	1.55995i	0.625808	−	0.779977i	$-0.284769\pi$
$992$		0			0
$993$	75.8924			2.40837
$994$		0			0
$995$	11.9854	+	8.32462i	0.379962	+	0.263908i
$996$		0			0
$997$			24.6133i			0.779512i	0.920918	+	0.389756i	$0.127441\pi$
$997$			24.6133i			0.779512i	−0.920918	+	0.389756i	$0.872559\pi$
$998$		0			0
$999$	−61.4672			−1.94474

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.n.b.559.4		40
4.3	odd	2		280.2.n.b.139.22	yes	40
5.4	even	2	inner	1120.2.n.b.559.39		40
7.6	odd	2	inner	1120.2.n.b.559.37		40
8.3	odd	2	inner	1120.2.n.b.559.3		40
8.5	even	2		280.2.n.b.139.18	yes	40
20.19	odd	2		280.2.n.b.139.19	yes	40
28.27	even	2		280.2.n.b.139.21	yes	40
35.34	odd	2	inner	1120.2.n.b.559.2		40
40.19	odd	2	inner	1120.2.n.b.559.40		40
40.29	even	2		280.2.n.b.139.23	yes	40
56.13	odd	2		280.2.n.b.139.17	✓	40
56.27	even	2	inner	1120.2.n.b.559.38		40
140.139	even	2		280.2.n.b.139.20	yes	40
280.69	odd	2		280.2.n.b.139.24	yes	40
280.139	even	2	inner	1120.2.n.b.559.1		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.n.b.139.17	✓	40	56.13	odd	2
280.2.n.b.139.18	yes	40	8.5	even	2
280.2.n.b.139.19	yes	40	20.19	odd	2
280.2.n.b.139.20	yes	40	140.139	even	2
280.2.n.b.139.21	yes	40	28.27	even	2
280.2.n.b.139.22	yes	40	4.3	odd	2
280.2.n.b.139.23	yes	40	40.29	even	2
280.2.n.b.139.24	yes	40	280.69	odd	2
1120.2.n.b.559.1		40	280.139	even	2	inner
1120.2.n.b.559.2		40	35.34	odd	2	inner
1120.2.n.b.559.3		40	8.3	odd	2	inner
1120.2.n.b.559.4		40	1.1	even	1	trivial
1120.2.n.b.559.37		40	7.6	odd	2	inner
1120.2.n.b.559.38		40	56.27	even	2	inner
1120.2.n.b.559.39		40	5.4	even	2	inner
1120.2.n.b.559.40		40	40.19	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.n (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.91053 q^{3} +(-1.83654 - 1.27559i) q^{5} +(-1.52252 + 2.16378i) q^{7} +5.47117 q^{9} +O(q^{10})\)	\(q-2.91053 q^{3} +(-1.83654 - 1.27559i) q^{5} +(-1.52252 + 2.16378i) q^{7} +5.47117 q^{9} +0.0929484 q^{11} +4.08979i q^{13} +(5.34529 + 3.71264i) q^{15} -4.24008 q^{17} +2.39835i q^{19} +(4.43134 - 6.29773i) q^{21} -4.52148 q^{23} +(1.74573 + 4.68534i) q^{25} -7.19241 q^{27} +4.35133i q^{29} +1.10465 q^{31} -0.270529 q^{33} +(5.55626 - 2.03174i) q^{35} +8.54612 q^{37} -11.9034i q^{39} -6.10466i q^{41} -4.60415i q^{43} +(-10.0480 - 6.97898i) q^{45} -7.93489i q^{47} +(-2.36386 - 6.58879i) q^{49} +12.3409 q^{51} -14.3740 q^{53} +(-0.170703 - 0.118564i) q^{55} -6.98047i q^{57} -10.6226i q^{59} -1.92656 q^{61} +(-8.32997 + 11.8384i) q^{63} +(5.21690 - 7.51105i) q^{65} -13.1636i q^{67} +13.1599 q^{69} +9.75994i q^{71} +6.19245 q^{73} +(-5.08100 - 13.6368i) q^{75} +(-0.141516 + 0.201119i) q^{77} +3.42089i q^{79} +4.52019 q^{81} +11.7848 q^{83} +(7.78705 + 5.40860i) q^{85} -12.6647i q^{87} -4.46220i q^{89} +(-8.84939 - 6.22679i) q^{91} -3.21511 q^{93} +(3.05932 - 4.40466i) q^{95} +10.1447 q^{97} +0.508536 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 40 q^{9}+O(q^{10}) \) 40 * q + 40 * q^9	\( 40 q + 40 q^{9} + 16 q^{25} - 16 q^{35} + 8 q^{49} + 32 q^{51} - 24 q^{65} - 72 q^{81} + 128 q^{99}+O(q^{100}) \) 40 * q + 40 * q^9 + 16 * q^25 - 16 * q^35 + 8 * q^49 + 32 * q^51 - 24 * q^65 - 72 * q^81 + 128 * q^99

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